Analysis of high-dimensional repeated measures designs: The one sample case

A one sample statistic is derived for the analysis of repeated measures design when the data are multivariate normal and the dimension, dd, can be large compared to the sample size, nn, i.e. d>nd>n. Quadratic and bilinear forms are used to define the statistic based on Box’s approximation [Box, G.E.P., 1954. Some theorems on quadratic forms applied in the study of analysis of variance problems I: Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics 25 (2), 290–302]. The statistic has an approximate χf2 distribution, even for moderately large nn. One of the main advantages of the statistic is that it can be used both for unstructured and factorially structured repeated measures designs. In the asymptotic derivations, it is assumed that n→∞n→∞ while dd remains finite and fixed. However, it is demonstrated through simulations that for nn as small as 10, the new statistic very closely approximates the target distribution, unaffected by even large values of d(d>n). The application is illustrated using a sleep lab example with n=10,d=18.